Lemma 2.4 ([1]}) If \((u_n,v_n)\rightharpoonup (u,v)\)  in \(H^1(\mathbb {R}^4,\mathbb {R}^2)\) , then up to a subsequence,
\(\int _{\mathbb {R}^4}|u_n|^2|v_n|^2-|u_n-u|^2|v_n-v|^2=\int _{\mathbb {R}^4}|u|^2|v|^2+o(1).\) 

